Edsko de Vries used to have a listing of the videos, but it is no longer available. After wresting a copy from a Google cache, I began working my way through the videos, but soon discovered that Edsko’s list was organized by subject, not topologically sorted. So I started making my own list, and have put it up here in the hopes that it may be useful to others. Suggestions, corrections, improvements, etc. are of course welcome!

As far as possible I have tried to arrange the order so that each video only depends on concepts from earlier ones. Along with each video you can also find my cryptic notes; I make no guarantee that they will be useful to anyone (even me!), but hopefully they will at least give you an idea of what is in each video. (For some of the earlier videos I didn’t take notes, so I have just copied the description from YouTube.)

I have a goal to watch two videos per week (at which rate it will take me about nine months to watch all of them); I will keep this list updated with new video links and notes as I go.

Natural transformations

Natural transformations 1

Alternative definition of natural transformation analogous to usual homotopy definition: a natural transformation is a functor where is the “categorical interval”, i.e. the two-object category with a single nontrivial morphism.

Monads 3A

The monad algebra law for (a triangle) just says that can’t do anything interesting on one-element lists: it has to just return the single element.

Identity and associativity laws for the monoid come from the other monad algebra law, saying how interacts with (a square), and from how the list functor is defined. We start with a way of mapping lists down to values, which bakes in the idea that it doesn’t matter how we associate the list.

Monads 4

Given two monad algebras and , a morphism between them consists of a morphism of underlying objects, , such that the obvious square commutes.

Example. List monad again. . A morphism of monoids is a function such that . See how this equation arises from the commuting square for monad morphisms, by starting with a 2-element list in upper left and following it around.

Given a particular mathematical theory, can it be expressed as the category of algebras for some monad? I.e. given a category , is it equivalent to for some ? (Answer: no, not in general, e.g. category of topological spaces can’t.)

But this is still an interesting question, more or less the question of “monadicity”. Category said to be monadic over category if can be expressed as category of algebras of monads over .

Adjunctions 5

Last time we showed every adjunction gives rise to a monad. What about the converse?

Answer: yes. In fact, given a monad, there is an entire category of adjunctions which give rise to it, which always has initial and terminal objects: these are the constructions found by Kleisli and by Eilenberg-Moore, respectively. Intuitively, any other adjunction giving rise to the monad can be described by the morphisms between it and the Kleisli and Eilenberg-Moore constructions.

What about ? Sends to the “free” -algebra on , with underlying set . Then evaluation map is . That is, . Need to check that this definition of really gives a monad algebra as a result. In this case the monad algebra laws are just the monad laws for !

Now define a unit and counit. is just the for the monad. is an algebra morphism from the free algebra on (i.e.) to : in fact, itself is such a morphism, by the second algebra law.

Intuition: this is the category of free algebras: is equivalent, under the adjunction, to , morphism between free algebras.

Note, for the Eilenberg-Moore category (last time) it was complicated to define the objects and simple to define the morphisms. For Kleisli, it’s the other way around. “Conservation of complicatedness.”

Adjunctions 7

The adjunction that comes from the Kleisli category, giving rise to the original monad .

Again, let be a monad. We will construct , where is the Kleisli category defined in Adjunctions 6, with .

sends objects to “free algebras”

Identity on objects.

On morphisms, sends to (equivalently ).

sends a “free algebra” to its “underlying object”

Sends to .

Sends to .

Unit and counit

we can take as the of the monad.

we can take to be id.

Adjunction laws come down to monad laws (left to viewer).

Given a monad on , we have a category of adjunctions giving rise to (morphisms are functors making everything commute). is the initial object and is terminal.

Question of monadicity: given an adjunction , is ? If so, say “ is monadic over ”, i.e. everything in can be expressed as monad algebras of . Or can say the adjunction is a “monadic adjunction”. Can also say that the right adjoint (forgetful functor ) “is monadic”. Monadic adjunctions are particularly nice/canonical.

String diagrams 2

Recall the interchange law, which says that vertical and horizontal composition of natural transformations commute. This guarantees that string diagrams are well-defined, since the diagram doesn’t specify which happens first.

Whiskering is represented in string diagrams by horizontally adjoining a straight vertical line.

String diagrams 3

Given an adjunction , we have natural transformations and , and two laws given by triangles. What do these look like as string diagrams? is a cap, a cup, and the triangle laws look like pulling wiggly strings straight!

String diagrams 5

Open-closed cobordisms

These are presented without any commentary or explanation that I can find. Each of the below videos just presents a 3D structure made out of pipe cleaners with no explanation. Maybe there is some other catsters video that presents a motivation or explanation for these; if I find it I will update the notes here. I can see that it might have something to do with string diagrams, and that you can make categories out of these sorts of topological structures (e.g. with gluing as composition) but otherwise I have no clue what this is about.

General limits and colimits 4

Formal definition of a limit: given a diagram , a limit for is an object together with a family of isomorphisms natural in . I.e. a natural correspondence between morphisms (the “factorization” from one cone to another) and morphisms (i.e. natural transformations) from to in the functor category (i.e. cones over with vertex ). That is, every cone with vertex has a unique factorization morphism, and vice versa!. The “vice versa” part is the surprising bit. If we have a limit then every morphism is the factorization for some cone to the universal cone.

If we set then etc. In particular corresponds to some cone, which is THE universal cone. The Yoneda lemma says (?) that the entire natural isomorphism is determined by this one piece of data (where goes). Note that both and are functors . The Yoneda lemma says that a natural transformation from to is isomorphic to — i.e. a cone with vertex , the universal cone.

General limits and colimits 6

Colimits using the same general formulation. “Just dualize everything”.

Cocone (“cone under the diagram”) is an object with morphisms from the objects in the diagram such that everything commutes.

Universal cocone: for any other cocone, there is a unique morphism from the universal cocone to the other cone which makes everything commute. Note it has to go that direction since the universal cocone is supposed to be a “factor” of other cocones.

In Eugenia’s opinion the word “cocone” is stupid.

More generally: natural isomorphism between cocones and morphisms. . Limits in are the same as colimits in , and vice versa.

All limits are terminal objects in a category of cones (and colimits are initial objects).

Since terminal objects are initial objects in (and vice versa), we can even say that all universal properties are initial objects (and terminal objects) somewhere.

Coequalisers 2

Quotient groups as coequalisers. Consider a group and a normal subgroup . In the category of groups, consider two parallel maps : the inclusion map , and the zero map which sends everything to the identity element . Claim: the coequaliser of these two maps is the quotient group , together with the quotient map .

Let’s see why. Suppose we have another group with a group homomorphism such that ; that is, for all . We must show there is a unique homomorphism which makes the diagram commute.

Notation: under the quotient map gets sent to ( iff ). For the homomorphism , send to . Note this is required to make things commute, which gives us uniqueness; we must check this is well-defined and a group homomorphism. If then . By definition, , and since is a group homomorphism, . Hence it is well-defined, and must additionally be a group homomorphism since and is a group homomorphism.

2-categories

2-categories 1

Generalization of categories: not just objects and morphisms, but also (2-)morphisms between the (1-)morphisms. Primordial example: categories, functors, and natural transformations.

Note: today, strict 2-categories, i.e. everything will hold on the nose rather than up to isomorphism. A bit immoral of us.

Recall: a (small) category is given by

A set of objects

for all , a set of morphisms

equipped with

identities: for all a function

composition: for all , a composition function .

unit and associativity laws.

To make this into a 2-category, we take the set of morphisms and categorify it. That turns some of the above functions into functors. Thus, a -category is given by a set of objects along with

a category for each

a functor for each

a composition functor.

etc.

(Note: why not turn the set of objects into a category? That’s a good question. Turns out we would get something different.)

Let’s unravel this a bit. If is a category then the objects are morphisms (of ) , and there can also be morphisms (of ) between these morphisms: -cells. -cells can be composed (“vertical” composition).

We also have the composition functor . On “objects” (which are -cells in ) the action of this functor is just the usual composition of -cells. On morphisms (i.e. -cells), it gives us “horiztonal” composition.

Next time: how functoriality gives us the interchange law.

2-categories 2

Interchange in a 2-category comes from functoriality of the composition functor. The key is to remain calm.

The functor is . On morphisms, it sends pairs of -cells to a single -cell, the horizontal composite. What does functoriality mean? It means if we have two (vertically!) composable pairs of -cells; the functor on their composition (i.e doing vertical composition pointwise) is the same as applying the functor to each (i.e. first doing the horizontal compositions) and then composing (vertically).

If we start with a monoid and consider the free group on its underlying elements, we can define a product using distributivity; so the free group on a monoid is a group. Formally, the free group monad lifts to the category of monoids (?).

Distributive laws 2

More abstract story behind our favorite example: combining a group and a monoid to get a ring.

Note: distributive law (at least in this example) is definitely non-invertible: you can turn a product of sums into a sum of products, but you can’t necessarily go in the other direction.

Main result: A distributive law is equivalent to a lift of to a monad on -. becomes a monad, and - is equivalent to -.

When is a monad? We need ; can do this if we have , then use . The laws for a distributive law ensure that this satisfies the monad laws.

Distributive law is equivalent to a lift of to a monad on -?

An -algebra looks like ; we want to send this to another -algebra, with carrier , i.e. some . But we have ; precomposing with gives us what we want, and the distributive law axioms ensure that is a monad on -.

- is equivalent to -?

Since is a monad on , a -algebra has an -algebra as its underlying object. So given some -algebra , a ’-algebra on it is a morphism of -algebras from to , that is,

This essentially says that an algebra for is simultaneously an algebra for and an algebra for which interact properly via the distributivity law.

An algebra for is . Clear that from a -algebra we get a algebra. What about vice versa? Just precompose with to get an -algebra, and with to get a -algebra. A -algebra says how to evaluate e.g. multiplication and additions all mixed up; precomposing with picks out the stuff with just multiplication or just addition. Apparently one can prove that the algebras you get this way do indeed interact nicely.

Distributive laws 3 (aka Monads 6)

Recall that a monad is a functor together with some natural transformations; we can see this as a construction in the -category of categories, functors, and natural transformations. We can carry out the same construction in any -category , giving monads in.

Let be a -category (e.g.). A monad in is given by

a -cell

a -cell

a pair of -cells and

satisfying the usual monad axioms.

In fact, we get an entire -category of monads inside !

What is a morphism of monads? A monad functor (i.e. a -cell in the -category of monads in ) is given by

a -cell

a -cell (Note, this is not backwards! This is what we will need to map algebras of the firs monad to algebras of the second.)

satisfying the axioms:

A monad transformation (i.e. a -cell in the -category of monads in ) is given by

a -cell , satisfying (something like that, see pasting diagrams of -cells in the video).

Distributive laws 4

Consider the -category of monads in an arbitrary -category ; monads in are distributive laws!

Recall that a monad in an arbitrary -category is a -cell equipped with an endo--cell and appropriate -cells and . In :

A -cell in , that is, a monad in .

A -cell , that is, a functor and .

-cells and . Can check that these turn into a monad.

Axioms on give exactly what is needed to make it a distributive law.

Summarizing more concisely/informally, a monad in is

A -cell

A pair of monads ,

A distributive law .

Consider the map . This actually defines an endofunctor on -, the category of (strict) -categories and (strict) -functors. In fact, Street showed that is a monad! The “monad monad”.

The multiplication has type . Recall that objects in are a pair of monads , plus a distributive law. In fact, the distributive law is precisely what is needed to make into a monad, which is the monad returned by the multiplication.

Group Objects and Hopf Algebras

Group Objects and Hopf Algebras 1

Take the idea of a group and develop it categorically, first in the category of sets and then transport it into other categories (though it may not be completely obvious what properties of we are using).

A group is of course a set with an associative binary product, inverses, and an identity element. Let’s make this categorical: don’t want to talk about internal structure of but just about as an object in .

So a group is:

an object

a multiplication morphism

an inverse morphism

a unit morphism (i.e. “universal element”)

together with axioms expressed as commutative diagrams:

(note to be pedantic we also need to use and )

where is the diagonal map (note the fact that we are using is the most interesting part; see forthcoming lectures) and is the unique map to a terminal set.

Group Objects and Hopf Algebras 3

The definition given last time won’t work in general for any monoidal category, but it does work for any Cartesian category (that is, monoidal categories where the monoidal operation is categorical product). Examples of Cartesian categories, in which it therefore makes sense to have group objects, include:

(category of topological spaces, with Cartesian product toplogy)

(cat. of smooth manifolds?)

(groups)

(categories)

Let’s see what a group object looks like in each of these examples.

In , a group object is a group.

In , a topological group.

In , a Lie group.

In , it turns out a group object is an Abelian group! (Details left as an exercise.)

In , we get a “crossed module”.

What about non-Cartesian monoidal categories? Simplest example is , category of (finite-dimensional) vector spaces with linear maps. Monoidal structure given by tensor product and complex numbers. Tensor product defined by

Group Objects and Hopf Algebras 4

We still want to be able to define group objects in monoidal categories which are not Cartesian.

Recall: if we have a monoidal category where is the categorical product, then every object is a comonoid in a unique way, and every morphism is a comonoid map.

Notation: in , an object with an associative binary operation and an identity is called a monoid; in it’s called an algebra. So when we generalize to arbitrary categories sometimes “monoid” is used, sometimes “algebra”.

A Hopf algebra is a group object in a general monoidal (tensor) category. Details next time.

Group Objects and Hopf Algebras 5

A Hopf algebra in a (braided) monoidal category is as follows. We don’t get comonoid stuff for free any more so we have to put it in “by hand”.

comonoid and

monoid and

“antipode” or inverse

(See video for string diagrams.) Note the monoid and comonoid also need to be “compatible”: this is where the braidedness comes in. In particular and need to be comonoid morphisms. So we need to be a coalgebra.

Lemma: suppose , are comonoids. Then is a coalgebra if the category is braided: using comonoid structures on and , and then using (associativity and) braiding we can flip inner around to get .

Can then write down what it means for to be a coalgebra map aka comonoid morphism; left as an exercise (or the next video).

Ends 2

That is, for every we have , such that . i.e. the family are the components of a natural transformation .

Note this goes in the other direction too, that is, a wedge is precisely the same thing as a function . Therefore, the universal such is precisely this set of natural transformations. (Can be thought of as “set of symmetries” of a category. Also the Hochschild cohomology.)

Ends 4

What happens if we combine these two results? First, look at the end from last time:

Let be a natural transformation on . That is, is a function , such that commutes with the underlying function of any equivariant map, i.e.

As we showed last time, for some .

Note is just a function , but has to commute with equivariant functions.

Now look at the end of the bare hom-functor in the category of -sets. i.e.

Now if , we have

What’s the difference? is now a family of equivariant maps. But note equivariant maps are determined by their underlying function. So any diagram of this form implies one of the previous form; the only thing we’ve added is that itself has to be equivariant (in the previous case could be any function). So in fact we have

.

i.e. we’re picking out some subset of . Question: which subset is it? That is, given such a we know for some ; which ’s can we get?

Consider the left-regular representation again. Then we know is just left-multiplication by some . But it has to commute with equivariant maps; picking the action on the particular element , this means for all

Adjunctions from morphisms 2

Baby examples of some particular adjunctions (in generality, they show up in Grothendieck’s 6 operations, Frobenius reciprocity, …). Idea: start with (e.g.) sets; to each set associate a category; to each morphism between sets we will get functors between the categories.

To the set associate the slice category .

Think of the objects of this slice category, , as “bundles over ”: a base space and a set above that, where each element of is associated to its fiber/preimage.

Another way to think of this is as a functor (considering as a discrete category), that picks out the fiber of each element of .

There is actually an equivalence of categories .

What about maps between sets? e.g.. As we’ll see, we get three associated maps , , and , with . Details in the next lecture.

Adjunctions from morphisms 3

Given , define the “pullback” of a bundle over to a bundle over written : to each we associate the fiber of . That is,

Now for the other direction. Idea: given a bundle over and , for each we have a set which are sent to that by ; we have to somehow combine their fibers to yield a fiber for . Several ways we could imagine doing it: disjoint union, product? In this case we want the product. That is,

Double Categories

Internal categories in . Recall that an internal category in is a pair of objects (representing objects) and (representing morphisms), and a pair of parallel arrows in recording the source and target of each morphism, all suitably equipped with unit and composition.

If and are themselves categories, and and are functors, then itself has sets of objects and morphisms with source and target functions, and the same for . Then the functors and have actions on morphisms and objects, so we get a square with two parallel arrows on each side.

are -cells.

are “vertical -cells”.

are “horizontal -cells”.

are -cells, which sit inside squares (not inside the area between two parallel -cells): each element of has corresponding sources and targets in both and , and the double commuting square described above ensures that the sources and targets of those have to match up in a square.

What about composition? Note and already come equipped with composition, which together give us “vertical composition” of -cells. Composition in the internal category gives horizontal composition of -cells.

Note if all vertical -cells are identities, this collapses to the usual idea of a -category. (Or symmetrically, with horizontal -cells being identities.)

Spans

Spans 1

NOTE: There seems to be no catsters video actually explaining what a “bicategory” is. According to the nlab it is a weaker version of a 2-category, where certain things are required to hold only up to coherent isomorphism rather than on the nose.

Let be a category with (chosen) pullbacks. is a bicategory with

-cells the objects of

-cells are spans .

-cells are morphisms between -cells, that is, spans. So a -cell between and is a morphism which makes things commute.

-cell composition: use pullback. Is this associative? Yes, up to isomorphism (because of universal property of pullbacks) but not on the nose. (Hence we get a bicategory and not a strict -category.)

Vertical -cell composition: just composition of the underlying morphisms.

Horizontal -cell composition: the two pullbacks induce a morphism between them.

Can check all the axioms etc.

Now, note monads can be constructed inside any bicategory, and are given by

a -cell

a -cell

-cells and satisfying the usual monad axioms (slightly modified to make them make sense)

It turns out that monads in are great! For example, monads in are small categories. Next time we’ll see why.

Spans 2

Monads in are small categories. These notes make a lot more sense when you can look at the diagrams. Watch the video or work out the diagrams yourself.

We have

a -cell, i.e. a set .

a -cell from to itself, i.e. a span (idea is that will be the set of morphisms, and and will pick out the source and target objects)

a -cell from (the boring span with all the same object and identity morphisms) to the -cell given above. This ends up being a function such that , that is, takes each object in to a morphism in having that object as both its source and target.

a -cell . is given by a pullback: a pair of morphisms such that the target of the first equals the source of the second, i.e. a composable pair. therefore has to take a composable pair and produce a single morphism in such that its source equals the source of the first and its target equals the target of the second.

And of course there are some monad laws which amount to the category laws.

Multicategories 2

We’ve seen a category of -spans, spans with a on the left. We’ll see that monads in - are multicategories.

Recall that is the list monad.

A monad in - is:

a set (which will represent objects of the multicategory)

a -cell is a -span, i.e. an object (representing morphisms of the multicategory) together with morphisms from to (picking out the sequence of input objects) and from to (picking the target object).

a -cell , representing the identity morphism with a single input (see video for a commutative diagrma)

a -cell which represents composition in a multicategory. See video for diagram!

Key point: we can actually do this with other monads ! And even on other categories with pullbacks, as long as preserves pullbacks (and and commutative diagrams are pullbacks). This yields a notion of a -multicategory. The source of each morphism is not just a list of objects but a -structure of objects.